Stillwater, November 2010

A Hybrid Genetic Algorithm for the Periodic Vehicle Routing Problem with Time Windows Michel Toulouse 1,2 Teodor Gabriel Crainic 2 Phuong Nguyen 2 1 Oklahoma State University 2 Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT) Stillwater, November 2010

Presentation Outline 1 Introduction 2 Periodic Vehicle Routing Problem with Time Windows 3 Genetic Algorithms 4 5 A Hybrid Genetic Algorithm for PVRPTW Conclusion 1

Introduction 2

Introduction 3

Vehicle Routing Problem (VRP) 4

VRP with Time Windows (VRPTW) 5

Description of PVRPTW The Periodic Vehicle Routing Problem with Time Windows (PVRPTW) is defined as having: A planning horizon of t days, n customers having a demand q i > 0, a service duration d i >0, a time window [e i, l i ], a service frequency f i and a set R i of allowable patterns of visit days, a single depot with time window [e 0, l 0 ], at which is based a fleet of m vehicles with limited on capacity and duration, a cost (or travel time) c ij > 0 between the locations. 6

Description of PVRPTW The PVRPTW aims to select a single visit day pattern per customer and design at most m vehicle routes on each day of the planning horizon such that: each route starts and ends at the depot in the interval [e 0, l 0 ], each customer i belongs to exactly f i routes over the horizon and is serviced in the interval [e i, l i ], the total capacity and duration of route k do not exceed Q k and D k, respectively, the total cost (or travel time) of all vehicles is minimized. 7

PVRPTW s mathematical formulation minimize subject to r R t T ( i, j) E k K i j V y ir c x t ij ijk = 1 i V (1) t t x = x i V, k K, t T (2) ijk j V jik t x = y a i V, t T (3) ijk ir rt C k K j V r R k K j V i, j S j V i V C C x t 0jk t 0 jk i m t T (4) t x S 1 S V, k K, t T (5) x ijk 1 k K, t T (6) t q x Q k K, t T i j V ijk t t t w w + s + c M(1 x ) (, i j) E, k K, t T (8) jk ik i ij ijk t e w l i V, k K, t T (9) i ik i C i V C x ( w + s + c ) D k K, t T (10) t t i0k ik i i0 8 C C (7)

Genetic Algorithms: Engineering view Two horses and a groom (Han Gan [Tang Dynasty]) 9

Genetic algorithms: Evolutionary view Species constantly have to adapt to changes in their environment. Fittest of a specie live long enough to breed (natural selection). They pass their genetic adaptive features to their offspring. Through several generations, the specie get the upper hand on environment changes. 10

GA as an optimization method A stochastic search method. The environment is the cost function. A specie is a set of solutions Individuals are solutions, the cost of a solution measure it fitness. Furthermore, solutions are encoded under a suitable representation. Individuals (solutions) are selected to breed based on a random procedure biased by the fitness of the solutions. Breeding consists to brake solutions into components and to reassemble components to create offspring. 11

GA as optimization method Generation of an initial population Parent population Select two parents Reproduction of a generation Generation of a new parent population from children population Reproduction of parents Mutation of children Children population NO Stop evolution? YES Get solution from children population 12

Pseudo code for generational GAs 13

Overview of our GA for PVRPTW START Initial population 1 st generation N th generation Fitter Random Fitter Unfit Roulette wheel Mating pool Crossover & mutation Offspring Repair New New population Immigration Children population NO N > genmax? STOP 14 YES

Representation of 15

Individual fitness The fitness function: fs ( ) = cs ( ) + αqs ( ) + βds ( ) + γw(s) where qs ( ), ds ( ), ws ( ): the total violation of capacity, duration and time windows, respectively, q d w α = h, β = h, γ = h q + d + w q + d + w q + d + w 2 2 2 2 2 2 2 2 2 h = cs ( worst) if no feasible solution cs ( bestfeasible) otherwise q, d, w are the violation of capacity, duration and time windows respectively averaged over the current population. 16

Algorithmic elements START Initial population 1 st generation N th generation Fitter Random Fitter Unfit Roulette wheel Mating pool Crossover & mutation Offspring Repair New New population Immigration Children population NO N > genmax? STOP 17 YES

The initial population Each customer is assigned a feasible pattern of visit days randomly. Solve VRPTW by applying: 1. Time-Oriented, Sweep Heuristic by Solomon. 2. Parallel route building by Potvin and Rousseau. 3. Our route construction method. 18

Roulette Wheel selection operator 20

The first crossover operator 22

The second crossover operator For each day t, one parent among {P 1, P 2 } is selected randomly, from which all routes in day t are copied into the offspring Off 2. Remove/insert customers from/into days such that the pattern of visit days of all customers are satisfied. 28

At issues GA not well adapted to constrained optimization, crossover create infeasible solutions. PVRPTW is a heavily constrained optimization problem. Usually repair strategies aim at regaining feasibility, but for PVRPTW this often leads to very poor solutions. Not only solutions are poor, but also their genetic make-up (building blocks hypothesis). Need to repair not only feasibility but also the building block features of the population. Make use of metaheuristics. 30

Repair strategies Phase 1: Simultaneously tackle routing and pattern improvements Unified Tabu of Cordeau et al. Random VNS of Pirkwieser and Raidl: order of neighborhood structures are chosen randomly Pattern improvement: explore all feasible patterns of all customers where each customer is reassigned to new pattern, one by one Phase 2: Routing improvements locally re-optimize the routes in which hybridized neighborhood structures with a set different route improvement techniques are used 31

Replacement npop parents are selected from the current population using Roulette wheel to build the mating pool and npop offspring are then created. The next generation is composed of the nkeep best among the pool of chromosomes in the current population (nkeep < npop) and npop new offspring. 33

Previous works 20 instances generated by Cordeau et al. 1. Unified Tabu Search of Cordeau et al. (2001) (1) move a customer, (2) change pattern of a customer, accept infeasible solutions. 2. Variable Neighborhood Search of Pirkwieser and Raidl (2008) change pattern of a customer, (2) move a segment, (3) exchange segments, accept worse solutions. 45 instances generated by Pirkwieser and Raidl 1. Hybrid scheme between VNS and ILP-based column generation approach (2009) 2. Multiple cooperating VNS (2010) 3. Hybrid scheme between multiple-vns and ILP-based column generation approach (2010) 34

Instances Benchmark #instances #customers #vehicles Planning period Cordeau et al. 20 [48, 288] [3, 20] 4 or 6 days S.Pirkwieser & Raidl 45 100 [7, 14] 4, 6 or 8 days 35

Experiment parameters Parameters Setting Final values Small instance Large instance Population size (npop) [50, 400] 100 350 Number of elite (nkeep) [25, 300] 60 200 Number of iterations applying UTB [20, 120] 60 100 Number of iterations applying VNS [100, 800] 100 200 36

Numerical Results Compare with currently best published results: For 20 instances generated by Cordeau et al.: produces 19 new best known solutions, with improved quality of 0.75% on average in term of best solutions cost. For 45 instances generated by S.Pirkwieser and Raidl: produces solutions with improve quality of 0.88% on average in term of average solutions cost. 37

Numerical Results Instances UTB VNS HGA %GAP to BKS No n T m D Q 1a 48 4 3 500 200 3007.84 2989.58 2989.58 0 2a 96 4 6 480 195 5328.33 5127.98 5107.51-0.40 3a 144 4 9 460 190 7397.10 7260.37 7158.77-1.40 4a 192 4 12 440 185 8376.95 8089.15 7981.85-1.33 5a 240 4 15 420 180 8967.90 8723.63 8666.59-0.65 6a 288 4 18 400 175 11686.91 11063.00 10999.90-0.57 7a 72 6 5 500 200 6991.54 6917.71 6892.71-0.36 8a 144 6 10 475 190 10045.05 9854.36 9751.66-1.04 9a 216 6 15 450 180 14294.97 13891.03 13707.30-1.32 10a 288 6 20 425 170 18609.72 18023.62 17754.20-1.49 1b 48 4 3 500 200 2318.37 2289.17 2284.83-0.19 2b 96 4 6 480 195 4276.13 4149.96 4141.15-0.21 3b 144 4 9 460 190 5702.07 5608.67 5567.15-0.74 4b 192 4 12 440 185 6789.73 6534.12 6471.74-0.95 5b 240 4 15 420 180 7102.36 6995.87 6963.11-0.47 6b 288 4 18 400 175 9180.15 8895.31 8855.97-0.44 7b 72 6 5 500 200 5606.08 5517.71 5509.08-0.16 8b 144 6 10 475 190 7987.64 7712.40 7677.68-0.45 9b 216 6 15 450 180 11089.91 10944.59 10874.80-0.64 10b 288 6 20 425 170 14207.64 14065.16 13851.40-1.52 38

Conclusion This algorithm outperforms the best existing methods for solving PVRPTW. It is part of a larger project to develop a cooperative system for VRP: A set S = {1, 2,,n} of n different search agents with dynamics x i (t + 1) = h i (x i (t - 1)), i S Agents are part of a network which can be represented by an oriented graph G = (V, E), V = {1, 2,,n} and E Vx V N = { j V (, i j) E} i Cooperation protocol: are the neighbors of search agent i x( t + 1) = f( ( x( t))) + hx ( ( t)) i j i j N i 39